Lecture 12: Specific Heat of Discrete Lattice

6.730 Physics for Solid State Applications

Outline

•Review Continuum Specific Heat Calculation •Density of Modes • Quantum Theory of Lattice Vibrations • Specific Heat for Lattice • Approximate Models March 1, 2004

Specific Heat of Solid How much energy is in each mode ?

Approach:

• Quantize the amplitude of vibration for each mode

• Treat each quanta of vibrational excitation as a bosonic particle, the phonon

• Use Bose-Einstein statistics to determine the number of phonons in each mode

1 Specific Heat Measurements

(hyperphysics.phy-astr.gsu.edu)

Hamiltonian for Discrete Lattice

Potential energy of bonds in 3-D lattice with basis:

For single atom basis in 3-D, µ & ν denote x,y, or z direction:

2 Hamiltonian for Discrete Lattice Plane Wave Expansion

The lattice wave can be represented as a superposition of plane waves (eigenmodes) with a known dispersion relation (eigenvalues)….

σ denotes polarization

Sum of harmonic oscillators for each mode

Simple Harmonic Oscillator

2 E 1 22 ψ ()x Ux( ) = 2 mω x

7 n = 3 E3 = 2 hω E 5 n = 2 2 = 2 hω E = 3 ω n = 1 1 2 h 1 n = 0 E0 = 2 hω x

3 Commutation Relation for Plane Wave Displacement

…commute unless we have same polarization and k-vector

Creation and Annhilation Operators for Lattice Waves

4 Operators for the Lattice Displacement

We will use this for electron-phonon scattering…

Specific Heat with Continuum Model for Solid

3-D continuum density of modes in dω :

5 Specific Heat with Discrete Lattice Density of Modes from Dispersion

1-D continuum density of modes in dω :

ωm

Specific Heat with Discrete Lattice Density of Modes from Dispersion

3-D continuum density of modes in dω :

6 Specific Heat of Solid How much energy is in each mode ?

Approach:

• Quantize the amplitude of vibration for each mode

• Treat each quanta of vibrational excitation as a bosonic particle, the phonon

• Use Bose-Einstein statistics to determine the number of phonons in each mode

Specific Heat of Solid

7 Approximate Models: The Debye Model Replace exact phonon dispersion relation and the density of modes with the density of modes of an elastic continuum, but include a cut-off frequency, the Debye frequency ωD

ωD

X Γ X

Determination of ωD The total number of modes of the system of N particles is 3N if there is one atom per primitive cell. The density of atoms per unit volume is then n = N/V. The total density of modes is then

Therefore,

Note: vs can be the “real” or “weighted” or “model” velocity; that is one has the freedom to choose the ratio ωD/vs so long as n is correct.

8 Specific Heat in the Deybe Model

Define the Debye Temperature as

Debye Specific Heat

C = 3nkB

C = AT3

θD = 102 K for Pb, 343 K for Cu, and 1860 K for C.

9 Einstein Model This is an even more simplistic model which approximates the

dispersion relation by one frequency ωE but demands that the number of modes per unit volume is correct.

ωD

ωE

X Γ X

Specific Heat in the Einstein Model

10 Combined Deybe and Einstein Models Use Debye mode for acoustic modes and Einstein for optical mode.

Optical modes ωE

ωD Acoustic modes

X Γ X ωD ωE